1 Gyrokinetic simulations of ETG turbulence and Gyrokinetic simulations of ETG turbulence and zonal flows in positive/reversed shear tokamaks zonal flows in positive/reversed shear tokamaks Yasuhiro Idomura Yasuhiro Idomura Japan Atomic Energy Research Institute Festival de Theorie Festival de Theorie 2005 2005 Aix- -en en- -Provance Provance, France, 4 , France, 4- -22 July 2005 22 July 2005 Aix Outline Gyrokinetic simulations of toroidal ETG turbulence � - Linear and quasi-linear analysis of ETG mode - ETG turbulence simulation in PS/RS tokamaks Self-organization in ETG turbulence �
2 Motivation to study ETG turbulence Motivation to study ETG turbulence � ETG turbulence is experimentally relevant candidate of χ e in tokamak – High suppression threshold ω ExB > γ than TEM (Stallard 1999) – Stiff T e profile consistent with critical L te of ETG (Hoang 2001) � Issues to be addressed Does ρ e scale ETG turbulence cause experimentaly relevant χ e ? – Yes: χ e ~10 χ GB ( χ GB = v te ρ te 2 / L te ) in ρ *-1 ~ ∞ ( ρ *-1 = a / ρ te ) local flux tube toroidal GK code (Jenko 2002) – No: χ e ~ χ GB in ρ *-1 ~100 global toroidal GF code (Labit 2003) What kind of structure formations does ETG turbulence show? – Streamers: positive shear flux tube toroidal GK code (Jenko 2002) – Zonal flows: reversed shear global slab GK code (Idomura 2000) � To examine these qualitatively and quantitatively different results, ETG turbulence is studied using global toroidal GK simulations – ρ * -dependence of toroidal ETG modes – Zonal flow and streamer formations in PS/RS-ETG turbulence
3 Basic equations Basic equations � Electrostatic gyrokinetic equation (Hahm 1988) 1 = + µ + φ 2 H m v B q e // e 2 g ∂ DF F { } = + = e e F , H 0 e ∂ Dt t ( ) d R c { } ≡ = + × ∇ φ + 2 ⋅ ∇ + µ ∇ R , H v b b q m v b b B ln B // e R e // R R * g dt q B e // ( ) * dv B { } ≡ = − ⋅ ∇ φ + ⋅ ∇ + µ ∇ 2 // v , H q m v b b B ln B // e R e // R R * dt m B g e // 2 cm v m v = + ∇ × µ = * ⊥ B B e // b , e R q 2 B e � Gyrokinetic Poisson equation ρ 2 1 ( [ ] ) ∫ − ∇ φ − ∇ ⋅ ∇ φ + φ = π δ δ + − 2 2 * 6 te 4 q F R ρ x m B d Z ⊥ ⊥ e e e e // λ λ 2 2 De Di
4 Calculation models of ETG turbulence simulation Calculation models of ETG turbulence simulation Electrostatic GK toroidal PIC code � Gyrokinetic electrons with adiabatic ions ( k ⊥ ρ ti >>1) � Annular wedge torus geometry � fixed B.C. φ = 0 – n = 0, N , 2 N … (N=25~100) – Quasi-ballooning representation � Global profile effects ( n e , T e , q , 1/ r ) � Self-consistent T e , n e are relaxed by heat/particle transport – ω * te -shearing effect – Reversed q ( r ) profile – Optimized particle loading � energy/particle conservation – Validity of simulation is checked by conservation properties !
5 High- - n n solver with quasi solver with quasi- -ballooning representation ballooning representation High = ∑ ( ) ( ) ( ) ( ) ( ) ( ) ˆ ˆ ˆ ˆ − ϕ + θ π φ θ ϕ φ θ ˆ φ = φ π ˆ in in q r i 2 n q r r , , r , e , r , 0 r , 2 e , r : reference surface s s n n n s n ˆ φ φ mode structure mode structure on the poloidal plane along the field line jump condition for periodicity in θ Realistic tokamak size a / ρ te ~10 4 : k θ ρ te ~1 ( q =1.4) m =5000 ~10 4 poloidal grids are needed without QB representation – ~10 2 poloidal grids are enough with QB representation –
6 Linear and quasi-linear analysis of ETG mode
7 Linear ETG growth rate spectrum Linear ETG growth rate spectrum Cyclone like parameters ( R 0 / L te =6.9, η e =3.12, a ~8600 ρ te ~150 ρ ti ) analysis domain – Unstable region spreads over n ~2000 ( m ~3000, k θ ρ te ~0.7) – RS-ETG mode is excited around q min surface (Idomura 2000) – Almost the same γ max in PS and RS configurations
8 Toroidal mode coupling in PS/RS configurations Toroidal mode coupling in PS/RS configurations � Positive shear configuration � Reversed shear configuration safety factor q safety factor q ballooning mode resonant perturbations stable mode ballooning mode q=(m+4)/n resonant perturbations q=(m+3)/n q=(m+2)/n q=(m+1)/n q=m/n q=(m-1)/n q=(m-2)/n nonresonant q=(m-3)/n slab mode q=(m-4)/n negative shear positive shear r r resonant surfaces resonant surfaces – Ballooning PS-ETG mode – Slab like RS-ETG mode – Big streamer structure in – Single helicity feature in weak field side weak shear region
9 * scan of ρ * scan of eigenfunctions eigenfunctions in PS/RS tokamaks in PS/RS tokamaks ρ � Positive shear configuration � Reversed shear configuration a / ρ te ~536 a / ρ te ~536 ~60 ρ te ~45 ρ te non-resonant resonant a / ρ te ~2146 a / ρ te ~2146 ∆ ρ ∝ ρ − * 1 / 2 r / te ~40 ρ te ~120 ρ te q min surface – ∆ r of PS-ETG mode is limited by ω * -shearing effect (Kim 1994) – ∆ r of RS-ETG mode is determined by q profile (Idomura 2000)
10 ρ * Mixing length theory and ρ -scaling scaling Mixing length theory and * - � Mixing length theory of ETG modes in PS/RS plasmas − − ∆ ρ ∝ ρ χ χ ∝ γ ρ – PS-ETG mode * 1 / 2 * 1 r / ML / te GB n ( ) − χ χ ∝ γ 1 / 2 – RS-ETG mode ∆ r ρ ∝ / L / L / L / L ML GB n ns n te ns n ( ) ( ) γ = γ − ′ ′ = 1 = 1 / 2 L / v L d ln n / dr L 2 qR / q r n n te n e ns 0 � ρ * scan of the saturation amplitude in single- n simulations Fixed local parameters R 0 / L te =6.9, η e =3.12 k θ ρ te ~0.3, a / R 0 =0.358 γ ≈ γ NL L γ NL : eddy turn over time – Small ρ * PS-ETG modes give order of magnitude higher saturation level than RS-ETG and large ρ * PS-ETG modes
11 ETG turbulence simulation in PS/RS tokamaks
12 Streamer formation in PS- -ETG turbulence ETG turbulence Streamer formation in PS � Linear phase ( t v te / L n ~110) � QL streamers ( t v te / L n ~175) weak field side θ ~ 0 k θ ρ te ~ 0.27 600 ρ te θ r � Saturation phase ( t v te / L n ~208) � Nonlinear streamers ( t v te / L n ~250) k θ ρ te ~ 0.17 ω ~ ω * e – PS-ETG turbulence is dominated by streamers – Streamers are characterized by ballooning structure and ω ~ ω * e
13 χ e in PS Extremely high χ in PS- -ETG turbulence ETG turbulence Extremely high e ( R 0 / L te ) crit ~4.5 R 0 / L te ~5.5 ~5 γ -1 R 0 / L te ~6.9 zonal flows R 0 / L te χ e ~10 χ GB χ e /( v te ρ te < V ExB >/ v te 2 / L te ) T e profile is strongly relaxed in a turbulent time scale ~5 γ -1
14 Convergence of saturation levels against wedge size Convergence of saturation levels against wedge size � Time history of fluctuation field energy – Saturation amplitude is converged against wedge torus size – Does nonlinear toroidal mode coupling (Lin 2004) lower saturation level?
15 Convergence of n n - -spectrum against wedge size spectrum against wedge size Convergence of � 1/100 wedge torus, 32 mode � 1/25 wedge torus, 128 mode | φ n | ( r / a ~0.5) second streamers k θ ρ te ~0.17 quasi-linear streamers k θ ρ te ~0.27 – Nonlinear spectrum is converged to coherent streamer mode – QL streamers are excited at linearly most unstable k θ ρ te – 2 nd streamers have coherent structure with k θ ρ te ~0.2 – Zonal flow component is very small
16 Zonal flow formation in RS- -ETG turbulence ETG turbulence Zonal flow formation in RS � Linear phase ( t v te / L n ~110) � Saturation phase ( t v te / L n ~207) weak field side θ ~ 0 k θ ρ te ~ 0.27 600 ρ te θ r q min � Secondary mode ( t v te / L n ~255) � Zonal flow formation ( t v te / L n ~380) – RS-ETG turbulence show qualitatively different behavior across q min – Zonal flows (streamers) appear in negative (positive) shear region
17 χ e χ gap structure in RS- -ETG turbulence ETG turbulence e gap structure in RS ( R 0 / L te ) crit ~3.7 zonal flows R 0 / L te q min quasi-steady χ e suppression zonal flows χ e /( v te ρ te < V ExB >/ v te 2 / L te ) q min q min T e gradient is sustained above its critical value in quasi-steady state
18 Summary(1) Summary(1) � ETG turbulence is studied using global toroidal GK simulations � Initial saturation levels consistent with the mixing length theory – Ballooning PS-ETG modes show Bohm like ρ * -scaling – Slab like RS-ETG modes show gyro-Bohm like ρ * -scaling – Small ρ * PS-ETG modes give an order of magnitude higher saturation level than RS-ETG and large ρ * PS-ETG modes � PS/RS ETG turbulences show different structure formations – PS-ETG turbulence is dominated by streamers � T e profile is quickly relaxed by large χ e ~10 χ GB – RS-ETG turbulence is characterized by zonal flows (streamers) in negative (positive) shear region � T e profile is sustained by χ e gap structure � These results suggest a stiffness of T e profile in PS tokamaks, and a possibility of the T e transport barrier in RS tokamaks
19 Self-organization in ETG turbulence
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